This is a slightly wacky sentence. It's not wrong - it does make sense and communicates the idea, it just forces you to do a bit of work to straighten it out in your head.

I think the biggest issue is the way they unnecessarily used present continuous tense with "be starting to get".

It's convoluted and adds syllables. You could eliminate the "be" and "to" entirely and change it to "start getting". That starts with an active verb which feels stronger and more natural.

So then it would be:

"This can't possibly be the same 9pm I used to start getting ready for a night out at".

That preserves the flow & punch of the delivery but shortens & simplifies it a lot without losing anything imo.

Also ending a sentence with a preposition can be awkward. You read "at" and you need to refer it back to 9pm near the start of the sentence. Plus it comes after another preposition, which itself is not acting as a preposition but as part of the nouned phrase "night out", so you end up with "out at". Again, not wrong, but it can be awkward. I think using "at which" can move it closer to the noun it's referring to but it's not necessarily better that way.

Make that change and it's, "This can't possibly be the same 9pm at which I used to start getting ready for a night out".

It's a little easier to parse, but honestly I think it loses something, because it doesn't have a casual delivery. "At which" is evidence that the sentence was very deliberately constructed. It adds a syllable and loses some punch. I'd stick with just the first change personally.

"This can't possibly be the same 9PM at which I once would prepare for outings."

Does this work better, or am I crazy?

And this is perfect example of using five words incorrectly, rather than the correct two.